Critical Set of the Master Function and Characteristic Variety of the Associated Gauss–Manin Differential Equations

Abstract

We consider a weighted family of n parallelly transported hyperplanes in a k-dimensional affine space and describe the characteristic variety of the Gauss–Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the associated point in the Grassmannian Gr(k, n). The Laurent polynomials are in involution. These statements generalize (Varchenko, Mathematics 2:218–231, 2014), where such a description was obtained for a weighted generic family of parallelly transported hyperplanes. An intermediate object between the differential equations and the characteristic variety is the algebra of functions on the critical set of the associated master function. We construct a linear isomorphism between the vector space of the Gauss–Manin differential equations and the algebra of functions. The isomorphism allows us to describe the characteristic variety. It also allowed us to define an integral structure on the vector space of the algebra and the associated (combinatorial) connection on the family of such algebras.